1879.] the Values of the Theta and Zeta Functions. 359 



1-2^ +% 16 -2 2 36 + &C. =f_±.fc / *(l • • ( 3 )> 



gl-jt-. a v+ 2 ¥ + g¥- *o. =_^J'»(l-i')* . . . (4), 



+ ^+ gr»+ 2*4+ &c. =^4: + . . . (5), 



7T 2 



2T V_^_ g« + 2 f*+ 2»- Ac. =^hKl-hy . . . (6), 



the signs of the terms in (4) and (6) being, after the first, alternately 

 negative and positive in pairs. 



Of these (3) is dedncible at once from 



l-2g + 2 2 4 -2 2 9 + &c. =00=—,— 



by the transformation of q into <f twice repeated. The same double 

 transformation converts (6) into (4), and (5) into the formula 



1 i 9 , 25 I 49 i _ <v 1 ^ 



so that (3), . . . (6) may be readily obtained independently of the 

 values of 0(JK), &c. 



If the modulus be changed from h to k' 9 (6) becomes 



77" 



where r = e~K^. 



Thus st-fl*-jV.+g¥ + j¥^ &c._ 1 ^KV 



t-ts— rnr — r ff -f- r^t + rri — &c. 2* 



so that if q, r be any quantities less than unity connected by the re- 

 lation 



log - log - = 7T S 

 q r 



then 



(log Vj(qi-ql-qV + q*i + ff V - &c.) 

 = ~ (log i J(r& -r^-rM + + rtt- &c.) : 



which may also be written 



